# Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

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Leibniz was born in Leipzig and completed his doctorate in philosophy at the age of 20 at the University of Altdorf. Throughout his life he engaged in scholarly work in several different fields. He was mainly self-taught in mathematics, since his interest in this subject developed when he was in his twenties. Leibniz arrived at the fundamental results of calculus independently, although a little later than Newton, but was the first to publish them, in 1684. Leibniz was very conscious of the power of good mathematical notation, and our notation for the derivative, dy/ dx, and the integral sign are due to him. He discovered the method of separation of variables (Section 2.2) in 1691, the reduction of homogeneous equations to separable ones (Section 2.2, Problem 30) in 1691, and the procedure for solving first order linear equations (Section 2.1) in 1694. He spent his life as ambassador and adviser to several German royal families, which permitted him to travel widely and to carry on an extensive correspondence with other mathematicians, especially the Bernoulli brothers. In the course of this correspondence many problems in differential equations were solved during the latter part of the seventeenth century.

The brothers Jakob (1654-1705) and Johann (1667-1748) Bernoulli of Basel did much to develop methods of solving differential equations and to extend the range of their applications. Jakob became professor of mathematics at Basel in 1687, and Johann was appointed to the same position upon his brother’s death in 1705. Both men were quarrelsome, jealous, and frequently embroiled in disputes, especially with each other. Nevertheless, both also made significant contributions to several areas of mathematics. With the aid of calculus they solved a number of problems in mechanics by formulating them as differential equations. For example, Jakob Bernoulli solved the differential equation Ó = [a3/(b2ó — a3)]1/2 in 1690 and in the same paper first used the term “integral” in the modern sense. In 1694 Johann Bernoulli was able to solve the equation dy/dx = ó/ ax. One problem to which both brothers contributed, and which led to much friction between them, was the brachistochrone problem (see

1.4 Historical Remarks

25

Problem 33 of Section 2.3). The brachistochrone problem was also solved by Leibniz and Newton in addition to the Bernoulli brothers. It is said, perhaps apocryphally, that Newton learned of the problem late in the afternoon of a tiring day at the Mint, and solved it that evening after dinner. He published the solution anonymously, but on seeing it, Johann Bernoulli exclaimed, “Ah, I know the lion by his paw.”

Daniel Bernoulli (1700-1782), son of Johann, migrated to St. Petersburg as a young man to join the newly established St. Petersburg Academy, but returned to Basel in 1733 as professor of botany, and later, of physics. His interests were primarily in partial differential equations and their applications. For instance, it is his name that is associated with the Bernoulli equation in fluid mechanics. He was also the first to encounter the functions that a century later became known as Bessel functions (Section 5.8).

The greatest mathematician of the eighteenth century, Leonhard Euler (1707-1783), grew up near Basel and was a student of Johann Bernoulli. He followed his friend Daniel Bernoulli to St. Petersburg in 1727. For the remainder of his life he was associated with the St. Petersburg Academy (1727-1741 and 1766-1783) and the Berlin Academy (1741-1766). Euler was the most prolific mathematician of all time; his collected works fill more than 70 large volumes. His interests ranged over all areas of mathematics and many fields of application. Even though he was blind during the last 17 years of his life, his work continued undiminished until the very day of his death. Of particular interest here is his formulation of problems in mechanics in mathematical language and his development of methods of solving these mathematical problems. Lagrange said of Euler’s work in mechanics, “The first great work in which analysis is applied to the science of movement.” Among other things, Euler identified the condition for exactness of first order differential equations (Section 2.6) in 1734-35, developed the theory of integrating factors (Section 2.6) in the same paper, and gave the general solution of homogeneous linear equations with constant coefficients (Sections 3.1,3.4,

3.5, and 4.2) in 1743. He extended the latter results to nonhomogeneous equations in 1750-51. Beginning about 1750, Euler made frequent use of power series (Chapter 5) in solving differential equations. He also proposed a numerical procedure (Sections

2.7 and 8.1) in 1768-69, made important contributions in partial differential equations, and gave the first systematic treatment of the calculus of variations.

Joseph-Louis Lagrange (1736-1813) became professor of mathematics in his native Turin at the age of 19. He succeeded Euler in the chair of mathematics at the Berlin Academy in 1766, and moved on to the Paris Academy in 1787. He is most famous for his monumental work Mecanique analytique, published in 1788, an elegant and comprehensive treatise of Newtonian mechanics. With respect to elementary differential equations, Lagrange showed in 1762-65 that the general solution of an nth order linear homogeneous differential equation is a linear combination of n independent solutions (Sections 3.2, 3.3, and 4.1). Later, in 1774-75, he gave a complete development of the method of variation of parameters (Sections 3.7 and 4.4). Lagrange is also known for fundamental work in partial differential equations and the calculus of variations.

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